Last modified: 2026-11-07 21:34:29 (PST)
A count outcome, such as the number of bike rentals in an hour, is not well described by a normal distribution when the counts are small, since the normal distribution puts probability below zero, and a count is never below zero. A Poisson regression model takes the event rate to be the exponential function of a linear function of the covariates.
Definition 1 (Poisson regression model) Let \(Y\) be a count outcome, \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{X}}\vphantom{X}}\) a covariate vector, and \(T\) the exposure magnitude over which the events in \(Y\) are counted. A Poisson regression model assumes that, given \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{X}}\vphantom{X}}= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\) and \(T = t\), \(Y\) has a Poisson distribution with mean \(\mu(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}, t) = \lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}) \cdot t\), where the event rate \(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\) has a log that is a linear function of the covariates:
\[ {\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}} = \beta_0 + \beta_1 x_1 + \ldots+ \beta_p x_p. \]
Definition 2 (Rate ratio) In a Poisson regression model, the rate ratio comparing covariate pattern \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\) to covariate pattern \(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}\) is the ratio of their event rates:
\[\text{ϙ}_{{\lambda}}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}: \mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}) \stackrel{\text{def}}{=}\frac{\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})}{\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})}.\]
Theorem 1 (Rate ratios in Poisson regression) In a Poisson regression model, suppose covariate \(x_j\) enters the linear function \({\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}}\) only through the single term \(\beta_j x_j\) (no interactions or transformations involving \(x_j\)). Let \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\) and \(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}\) be two covariate patterns that are identical except that \(x_j = a\) and \(x^*_j = b\). Then
\[\text{ϙ}_{{\lambda}}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}: \mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}) = {\operatorname{exp}\mathopen{}\left\{\beta_j (a - b)\right\}\mathclose{}} \tag{1}\]
and for any common exposure magnitude \(t > 0\), the ratio of the means is the same rate ratio:
\[\frac{\mu(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}, t)}{\mu(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}, t)} = \text{ϙ}_{{\lambda}}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}: \mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})\]
Proof. Because \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\) and \(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}\) agree on every covariate except \(x_j\), every term of \({\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}}\) except \(\beta_j x_j\) is shared with \({\ln\mathopen{}\left(\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})\right)\mathclose{}}\):
\[ \begin{aligned} {\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}} - {\ln\mathopen{}\left(\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})\right)\mathclose{}} &= \mathopen{}\left(\beta_0 + \beta_1 x_1 + \ldots+ {\color{red}{\beta_j a}} + \ldots+ \beta_p x_p\right)\mathclose{} \\ &\phantom{={}} - \mathopen{}\left(\beta_0 + \beta_1 x_1 + \ldots+ {\color{red}{\beta_j b}} + \ldots+ \beta_p x_p\right)\mathclose{} && \text{(substitute the linear function)} \\ &= {\color{red}{\beta_j a}} - {\color{red}{\beta_j b}} && \text{(cancel the shared terms)} \\ &= {\color{red}{\beta_j (a - b)}} && \text{(factor out } \beta_j \text{)} \end{aligned} \]
Then the rate ratio is:
\[ \begin{aligned} \text{ϙ}_{{\lambda}}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}: \mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}) &= \frac{\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})}{\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})} && \text{(definition of rate ratio)} \\ &= \frac{{\operatorname{exp}\mathopen{}\left\{{\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}}\right\}\mathclose{}}}{{\operatorname{exp}\mathopen{}\left\{{\ln\mathopen{}\left(\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})\right)\mathclose{}}\right\}\mathclose{}}} && \text{(exponential undoes the log)} \\ &= {\operatorname{exp}\mathopen{}\left\{{\ln\mathopen{}\left(\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})\right)\mathclose{}} - {\ln\mathopen{}\left(\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})\right)\mathclose{}}\right\}\mathclose{}} && \text{(quotient rule for exponentials)} \\ &= {\operatorname{exp}\mathopen{}\left\{\beta_j (a - b)\right\}\mathclose{}} && \text{(substitute the difference of linear functions)} \end{aligned} \]
For the means at a common exposure \(t > 0\):
\[ \begin{aligned} \frac{\mu(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}, t)}{\mu(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}, t)} &= \frac{\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}) \cdot t}{\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}) \cdot t} && \text{(Poisson regression mean function)} \\ &= \frac{\lambda(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}})}{\lambda(\mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}})} && \text{(cancel } t > 0 \text{)} \\ &= \text{ϙ}_{{\lambda}}(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}: \mathord{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*}}) && \text{(definition of rate ratio)} \end{aligned} \]
Example 1 (Rate ratio for a ten-year age difference)
Suppose a Poisson regression model for emergency-room visits includes age in years, \(x_1\), with coefficient \(\beta_1 = 0.03\) and no interactions involving age. By Theorem 1 with \(a = 60\) and \(b = 50\), the rate ratio comparing a 60-year-old to a 50-year-old with the same values of every other covariate is:
\[ \begin{aligned} \text{ϙ}_{{\lambda}}(x_1 = 60 : x_1 = 50) &= {\operatorname{exp}\mathopen{}\left\{\beta_1 (60 - 50)\right\}\mathclose{}} && \text{(rate ratio theorem)} \\ &= {\operatorname{exp}\mathopen{}\left\{0.03 \cdot 10\right\}\mathclose{}} && \text{(substitute } \beta_1 \text{ and subtract)} \\ &= {\operatorname{exp}\mathopen{}\left\{0.3\right\}\mathclose{}} && \text{(multiply)} \\ &\approx 1.350 && \text{(evaluate)} \end{aligned} \]
A one-year age difference instead gives \({\operatorname{exp}\mathopen{}\left\{0.03\right\}\mathclose{}} \approx 1.030\), a 3% higher event rate per year of age. The ten-year rate ratio is the one-year rate ratio multiplied by itself ten times, \({\operatorname{exp}\mathopen{}\left\{0.3\right\}\mathclose{}} = \mathopen{}\left({\operatorname{exp}\mathopen{}\left\{0.03\right\}\mathclose{}}\right)\mathclose{}^{10}\), because rate ratios multiply across successive one-year steps.